Discover the beauty of extreme Venn diagrams

If you think Venn diagrams are just a bunch of interlocking circles, think again. Pushing this iconic branch of mathematics to its limits reveals just how varied – and beautiful – these diagrams can be. This gallery showcases some of the wilder possibilities, including the most recent breakthrough in Venn geometry – the first simple, symmetric diagram to encompass a whopping 11 sets. Jacob Aron

Venn diagrams typically use overlapping circles to show all possible relationships between sets. Traditionally, they include just two or three sets. This example shows how red, blue and green lights produce other colours when they overlap.

Venn diagrams get tricky when they include more than three sets – here, a fourth set has to be stretched across the other three.

(Image: Kopophex/Wikimedia Commons)

The stretching method starts to fall apart once you reach six sets. The sets start to layer back and forth, and you end up with the kind of geometric gymnastics shown here.

(Image: Kopophex/Wikimedia Commons)

There are more aesthetic ways of illustrating four-set Venn diagrams, such as this example created by John Venn himself, who devised the first diagrams in the 1880s. However, his example uses overlapping ellipses and doesn't have the same symmetry as two and three sets.

If you are after the pleasing, rotational symmetry of the typical two- and three-set diagrams, however, mathematicians have proved that you must have a prime number of sets. Here five ellipses overlap to form a symmetrical Venn diagram that was first created by Croatian mathematician Branko Grünbaum in 1975.

( Image: Wikimedia Commons)

Things get prettier as the number of sets increases: Grünbaum and Edwards independently discovered this symmetrical Venn diagram for seven sets, the next prime number.

(Image: Frank Ruskey and Mark Weston)

The next prime number after 7 is 11, but it was tough to find a simple, symmetric diagram for 11 sets. The best that mathematicians could do was this complex, overlapping diagram discovered by US mathematician Peter Hamburger.

(Image: Frank Ruskey and Mark Weston)

Things changed when Khalegh Mamakani and Frank Ruskey at the University of Victoria in British Columbia, Canada, discovered the diagram shown here. Named Newroz, it is the first simple, symmetric 11-set Venn diagram.